Reduction of the Hurwitz space of metacyclic covers

The goal of this talk is to describe the reduction of the Hurwitz space parameterizing Galois covers of P^1 branched of order prime-to-p at four points, with Galois group $G=\ZZ/p\rtimes \ZZ/m$. Here m divides p-1. There are three possibilities for the reduction of a connected component H of the Hurwitz space, depending on whether the G-covers parameterized by H have (a) all good reduction, (b) all bad reduction, (c) or whether some have good and some have bad reduction. The description of the reduction of H in Case (c) generalizes the reduction of the modular curve X_1(p). An important tool in the proof is the definition of an analog of the Hasse invariant and show that it satisfies a certain hypergeometric differential equation.